• Almgren, A., J. Bell, P. Colella, L. Howell, and M. Welcome, 1998: A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comput. Phys., 142 , 146.

    • Search Google Scholar
    • Export Citation
  • Arakawa, A., and V. Lamb, 1977: Computational design of the basic dynamical process of the UCLA GCM. Methods in Computational Physics, J. Chang, Ed., Academic Press, 173–265.

    • Search Google Scholar
    • Export Citation
  • Arakawa, A., and V. Lamb, 1981: A potential enstrophy and energy conserving scheme for the shallow water equations. Mon. Wea. Rev., 109 , 1836.

    • Search Google Scholar
    • Export Citation
  • Augenbaum, J., and C. Peskin, 1985: On the construction of the Voronoi mesh on the sphere. J. Comput. Phys., 59 , 177192.

  • Baumgardner, J., and P. Frederickson, 1985: Icosahedral discretization of the two-sphere. SIAM J. Sci. Comput., 22 , 11071115.

  • Berger, M., and P. Colella, 1989: Local adaptive grid refinement for shock hydrodynamics. J. Comput. Phys., 82 , 6484.

  • Bonaventura, L., 2000: A semi-implicit, semi-Lagrangian scheme using the height coordinate for a nonhydrostatic and fully elastic model of atmospheric flows. J. Comput. Phys., 158 , 186213.

    • Search Google Scholar
    • Export Citation
  • Bonaventura, L., 2003: Development of the ICON dynamical core: Modelling strategies and preliminary results. Proc. ECMWF/SPARC Workshop on Modelling and Assimilation for the Stratosphere and Tropopause, Reading, United Kingdom, ECMWF, 197–214.

  • Bonaventura, L., and G. Rosatti, 2002: A cascadic conjugate gradient algorithm for mass conservative, semi-implicit discretization of the shallow water equations on locally refined structured grids. Int. J. Numer. Methods Fluids, 40 , 217230.

    • Search Google Scholar
    • Export Citation
  • Bracco, A., J. McWilliams, G. Murante, A. Provenzale, and J. Weiss, 2000: Revisiting freely decaying two-dimensional turbulence at millennial resolution. Phys. Fluids, 12 , 29312941.

    • Search Google Scholar
    • Export Citation
  • Casulli, V., and P. Zanolli, 1998: A three-dimensional semi-implicit algorithm for environmental flows on unstructured grids. Proceedings of Numerical Methods for Fluid Dynamics VI, M. J. Baines, Ed., ICFD Oxford University Computing Laboratory, 57–70.

    • Search Google Scholar
    • Export Citation
  • Casulli, V., and R. Walters, 2000: An unstructured grid, three-dimensional model based on the shallow water equations. Int. J. Numer. Methods Fluids, 32 , 331348.

    • Search Google Scholar
    • Export Citation
  • Chorin, A., and J. Marsden, 1993: A Mathematical Introduction to Fluid Mechanics. 3d ed Springer.

  • Dukowicz, J., 1995: Mesh effects for Rossby waves. J. Comput. Phys., 119 , 188194.

  • Gavrilov, M., 2004: On nonstaggered rectangular grids using streamfunction and velocity potential or vorticity and divergence. Mon. Wea. Rev., 132 , 15181521.

    • Search Google Scholar
    • Export Citation
  • Gill, A., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Giraldo, F. X., 2000: Lagrange–Galerkin methods on spherical geodesic grids: The shallow water equations. J. Comput. Phys., 160 , 336368.

    • Search Google Scholar
    • Export Citation
  • Girault, V., and P. Raviart, 1986: Finite Element Methods for the Navier–Stokes Equations. Lecture Notes in Mathematics, Springer-Verlag.

    • Search Google Scholar
    • Export Citation
  • Gross, E., L. Bonaventura, and G. Rosatti, 2002: Consistency with continuity in conservative advection schemes for free-surface models. Int. J. Numer. Methods Fluids, 38 , 307327.

    • Search Google Scholar
    • Export Citation
  • Harlow, F., and J. Welch, 1965: Numerical calculation of time dependent viscous incompressible flow. Phys. Fluids, 8 , 21822189.

  • Heikes, R., and D. Randall, 1995a: Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part I: Basic design and results of tests. Mon. Wea. Rev., 123 , 18621880.

    • Search Google Scholar
    • Export Citation
  • Heikes, R., and D. Randall, 1995b: Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part II: A detailed description of the grid and an analysis of numerical accuracy. Mon. Wea. Rev., 123 , 18811887.

    • Search Google Scholar
    • Export Citation
  • Hermeline, F., 1993: Two coupled particle-finite volume methods using Delaunay–Voronoi meshes for the approximation of Vlasov–Poisson and Vlasov–Maxwell equations. J. Comput. Phys., 106 , 118.

    • Search Google Scholar
    • Export Citation
  • Hoskins, B., 1973: Stability of the Rossby–Haurwitz wave. Quart. J. Roy. Meteor. Soc., 99 , 723745.

  • Jakob-Chien, R., J. Hack, and D. Williamson, 1995: Spectral transform solutions to the shallow water test set. J. Comput. Phys., 119 , 164187.

    • Search Google Scholar
    • Export Citation
  • Janjic, Z., 1984: Nonlinear advection schemes and energy cascade on semi-staggered grids. Mon. Wea. Rev., 111 , 12341245.

  • Jöckel, P., R. von Kuhlmann, M. Lawrence, B. Steil, C. Brenninkmeijer, P. Crutzen, P. Rasch, and B. Eaton, 2001: On a fundamental problem in implementing flux-form advection schemes for tracer transport in 3-dimensional general circulation and chemistry transport models. Quart. J. Roy. Meteor. Soc., 127 , 10351052.

    • Search Google Scholar
    • Export Citation
  • Le Roux, D. L., A. Staniforth, and C. Lin, 1998: Finite elements for shallow-water equation ocean models. Mon. Wea. Rev., 126 , 19311951.

    • Search Google Scholar
    • Export Citation
  • Leveque, R., 1996: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Sci. Comput., 33 , 627665.

  • Lin, S., and R. Rood, 1997: An explicit flux-form semi-Lagrangian shallow water model on the sphere. Quart. J. Roy. Meteor. Soc., 123 , 24772498.

    • Search Google Scholar
    • Export Citation
  • Liu, X., 1993: A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws. SIAM J. Numer. Anal., 30 , 701716.

    • Search Google Scholar
    • Export Citation
  • Majewski, D., and Coauthors, 2002: The operational global icosahedral–hexagonal gridpoint model GME: Description and high-resolution tests. Mon. Wea. Rev., 130 , 319338.

    • Search Google Scholar
    • Export Citation
  • Massey, W., 1977: Algebraic Topology: An Introduction. Springer Verlag, 261 pp.

  • Mesinger, F., 1981: Horizontal advection schemes on a staggered grid: An enstrophy and energy conserving model. Mon. Wea. Rev., 109 , 467478.

    • Search Google Scholar
    • Export Citation
  • Mesinger, F., and A. Arakawa, 1976: Numerical Methods Used in Atmospheric Models. Vol. I, GARP Publication Series, No. 17, WMO.

  • Morton, K., and P. Roe, 2001: Vorticity preserving Lax–Wendroff type schemes for the system wave equation. SIAM J. Sci. Comput., 23 , 170192.

    • Search Google Scholar
    • Export Citation
  • Ničković, S., 1994: On the use of hexagonal grids for simulation of atmospheric processes. Contrib. Atmos. Phys., 67 , 2,. 103107.

  • Ničković, S., M. Gavrilov, and I. Tošić, 2002: Geostrophic adjustment on hexagonal grids. Mon. Wea. Rev., 130 , 668683.

  • Nicolaides, R., 1992: Direct discretization of planar div-curl problems. SIAM J. Numer. Anal., 29 , 3256.

  • Pedlosky, J., 1987: Geophysical Fluid Dynamics. Springer Verlag, 710 pp.

  • Quarteroni, A., and A. Valli, 1994: Numerical Approximation of Partial Differential Equations. Springer-Verlag.

  • Quiang, D., M. Gunzburger, and J. Lili, 2003: Voronoi-based finite volume methods, optimal Voronoi meshes and PDEs on the sphere. Comput. Methods Appl. Mech. Eng., 192 , 39333957.

    • Search Google Scholar
    • Export Citation
  • Randall, D., 1994: Geostrophic adjustment and the finite-difference shallow-water equations. Mon. Wea. Rev., 122 , 13711377.

  • Raviart, P., and J. Thomas, 1977: A mixed finite element method for 2nd order elliptic problems. Mathematical Aspects of Finite Element Methods, I. Galligani and E. Magenes, Eds., Lecture Notes in Mathematics, Springer-Verlag, 292–315.

    • Search Google Scholar
    • Export Citation
  • Rebay, S., 1993: Efficient unstructured mesh generation by means of Delaunay triangulation and Bowyer–Watson algorithm. J. Comput. Phys., 106 , 125138.

    • Search Google Scholar
    • Export Citation
  • Ringler, T., and D. Randall, 2002: A potential enstrophy and energy conserving numerical scheme for solution of the shallow-water equations on a geodesic grid. Mon. Wea. Rev., 130 , 13971410.

    • Search Google Scholar
    • Export Citation
  • Ringler, T., R. Heikes, and D. Randall, 2000: Modeling the atmospheric general circulation using a spherical geodesic grid: A new class of dynamical cores. Mon. Wea. Rev., 128 , 24712490.

    • Search Google Scholar
    • Export Citation
  • Sadourny, R., 1969: Numerical integration of the primitive equations on a spherical grid with hexagonal cells. Proc. WMO/IUGG NWP Symp., Tokyo, Japan, Japan Meteorological Agency, 45–52.

  • Sadourny, R., 1975: The dynamics of finite-difference models of the shallow-water equations. J. Atmos. Sci., 32 , 680689.

  • Sadourny, R., and P. Morel, 1969: A finite-difference approximation of the primitive equations for a hexagonal grid on a plane. Mon. Wea. Rev., 97 , 439445.

    • Search Google Scholar
    • Export Citation
  • Sadourny, R., A. Arakawa, and Y. Mintz, 1968: Integration of the nondivergent barotropic vorticity equation with an icosahedral-hexagonal grid for the sphere. Mon. Wea. Rev., 96 , 351356.

    • Search Google Scholar
    • Export Citation
  • Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 378 pp.

  • Schär, C., and P. Smolarkiewicz, 1996: A synchronous and iterative flux-correction formalism for coupled transport. J. Comput. Phys., 128 , 101120.

    • Search Google Scholar
    • Export Citation
  • Smith, K., G. Boccaletti, C. Henning, I. Marinov, C. Tam, I. Held, and G. Vallis, 2002: Turbulent diffusion in the geostrophic inverse cascade. J. Fluid Mech., 469 , 1348.

    • Search Google Scholar
    • Export Citation
  • Stuhne, G., and W. Peltier, 1999: New icosahedral grid-point discretizations of the shallow water equations on the sphere. J. Comput. Phys., 148 , 2358.

    • Search Google Scholar
    • Export Citation
  • Thuburn, J., 1997: A PV-based shallow-water model on a hexagonal–icosahedral grid. Mon. Wea. Rev., 125 , 23282347.

  • Thuburn, J., and Y. Li, 2000: Numerical simulation of Rossby–Haurwitz waves. Tellus, 52A , 181189.

  • Tomita, H., M. Tsugawa, M. Satoh, and K. Goto, 2001: Shallow water model on a modified icosahedral grid by using spring dynamics. J. Comput. Phys., 174 , 579613.

    • Search Google Scholar
    • Export Citation
  • Williamson, D., 1968: Integration of the barotropic vorticity equation on a spherical geodesic grid. Tellus, 20 , 642653.

  • Williamson, D., 1979: Difference approximations for fluid flow on a sphere. Numerical Methods Used in Atmospheric Models, Vol. II, GARP Publication Series, No. 17, WMO, 53–123.

  • Williamson, D., J. Drake, J. Hack, R. Jakob, and R. Swarztrauber, 1992: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys., 102 , 221224.

    • Search Google Scholar
    • Export Citation
  • Winninghoff, F., 1968: On the adjustment toward a geostrophic balance in a simple primitive equation model. Ph.D. thesis, University of California, Los Angeles.

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Analysis of Discrete Shallow-Water Models on Geodesic Delaunay Grids with C-Type Staggering

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  • 1 Max-Planck-Institut für Meteorologie, Hamburg, Germany
  • | 2 Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
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Abstract

The properties of C-grid staggered spatial discretizations of the shallow-water equations on regular Delaunay triangulations on the sphere are analyzed. Mass-conserving schemes that also conserve either energy or potential enstrophy are derived, and their features are analogous to those of the C-grid staggered schemes on quadrilateral grids. Results of numerical tests carried out with explicit and semi-implicit time discretizations show that the potential-enstrophy-conserving scheme is able to reproduce correctly the main features of large-scale atmospheric motion and that power spectra for energy and potential enstrophy obtained in long model integrations display a qualitative behavior similar to that predicted by the decaying turbulence theory for the continuous system.

Corresponding author address: Luca Bonaventura, Max-Planck-Institut für Meteorologie, Bundesstraße 53, 20146, Hamburg, Germany. Email: bonaventura@dkrz.de

Abstract

The properties of C-grid staggered spatial discretizations of the shallow-water equations on regular Delaunay triangulations on the sphere are analyzed. Mass-conserving schemes that also conserve either energy or potential enstrophy are derived, and their features are analogous to those of the C-grid staggered schemes on quadrilateral grids. Results of numerical tests carried out with explicit and semi-implicit time discretizations show that the potential-enstrophy-conserving scheme is able to reproduce correctly the main features of large-scale atmospheric motion and that power spectra for energy and potential enstrophy obtained in long model integrations display a qualitative behavior similar to that predicted by the decaying turbulence theory for the continuous system.

Corresponding author address: Luca Bonaventura, Max-Planck-Institut für Meteorologie, Bundesstraße 53, 20146, Hamburg, Germany. Email: bonaventura@dkrz.de

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